Symplectic areas, quantization, and dynamics in electromagnetic fields

Abstract

A gauge invariant quantization in a closed integral form is developed over a linear phase space endowed with an inhomogeneous Faraday electromagnetic tensor. An analog of the Groenewold product formula (corresponding to Weyl ordering) is obtained via a membrane magnetic area, and extended to the product of N symbols. The problem of ordering in quantization is related to different configurations of membranes: a choice of configuration determines a phase factor that fixes the ordering and controls a symplectic groupoid structure on the secondary phase space. A gauge invariant solution of the quantum evolution problem for a charged particle in an electromagnetic field is represented in an exact continual form and in the semiclassical approximation via the area of dynamical membranes.

39 pages, 17 figures

References in corpus (3)

• Product rule for gauge invariant Weyl symbols and its application to the semiclassical description of guiding center motion
• Connes' Tangent Groupoid and Strict Quantization
• Quantization of a particle in a background Yang-Mills field

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• Magnetic pseudodifferential operators with coefficients in C*-algebras
• Strict Deformation Quantization for a Particle in a Magnetic Field
• On the regularity of the Hausdorff distance between spectra of perturbed magnetic Hamiltonians
• Heisenberg Evolution WKB and Symplectic Area Phases
• Cotangent bundle quantization: Entangling of metric and magnetic field
• Two-parameter Asymptotics in Magnetic Weyl Calculus
• Magnetic pseudodifferential operators represented as generalized Hofstadter-like matrices
• Dirac's magnetic monopole and the Kontsevich star product
• Estimating the number of negative eigenvalues of a relativistic Hamiltonian with regular magnetic field
• Application of symbol-operator technique for effective action computation
• On Fréchet-Hilbert Algebras
• Magnetic Fourier Integral Operators
• Intrinsic Dynamics of Symplectic Manifolds: Membrane Representation and Phase Product
• Positive Quantization in the Presence of a Variable Magnetic Field
• Eigenfunctions decay for magnetic pseudodifferential operators
• Abstract composition laws and their modulation spaces
• The Modulation Mapping for Magnetic Symbols and Operators
• Gradient expansion of the non-Abelian gauge-covariant Moyal star-product
• Flashes of noncommutativity
• Unicity of the integrated density of states for relativistic Schroedinger operators with regular fields and singular electric potentials